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I remember reading somewhere that in ancient times they were not treating a ratio like a division as we do.
I was wondering is there a subtle distinction between the concept of the ratio and the idea of division that made them cautious about this?

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  • $\begingroup$ in ancient times they did mostly straight edge compass geometry style questions. In explicitly geometry with ratios allowing for similar objects it is less clear with division $\endgroup$ May 17, 2016 at 17:51
  • $\begingroup$ But why they were not willing to use division for a ratio? $\endgroup$
    – Jim
    May 17, 2016 at 17:53
  • $\begingroup$ they also had less of a formal definition of a number, and more of a given unit length. in short ratios were fat easier to understand without the existence of numbers. I.e. this stick when compared with this stick is similar to this needle when compared to this needle. it was difficult to prove that this was always devision at first glance where division is the inverse operator of multiplication. Also ratios don't allow for no stick i.e. an invalid comparison. much easier then visualizing the same for 0 and division. $\endgroup$ May 17, 2016 at 17:59
  • $\begingroup$ See this post for some comments on the ancient Greek math concepts of number and ratio. $\endgroup$ May 17, 2016 at 19:37

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They did not work so much with either ratio or division as with proportion. This was denoted $a:b::c:d$. For example, altitudes $a$ and $b$ will be in the same proportion as the corresponding triangles with the same base. Today we would write this as $\frac{a}{b}=\frac{c}{d}$ but for the Greeks the only numbers were natural and proportion was a relation among actual entities rather than abstract numbers.

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